Bivariate Distribution

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A bivariate distribution gives probabilities for outcomes of two random variables occurring at the same time.

More precisely, it is a discrete joint distribution with two variables of interest, usually denoted X and Y. A “discrete joint distribution” has a finite set of possible outcomes (for example, 2, 3, or 99), whereas for the continuous version — the continuous joint distribution — you can describe relationships between an infinite number of variables. Continuous joint distributions can be described by a non-negative function [1].

Bivariate Distribution Table

Each bivariate distribution table is unique; its look depends on what variables you are studying. A table of rows and columns can be used to display discrete random variables with a finite (fixed) number of values. The rows represent one of the variables, the columns the other. For example, the following table shows the bivariate distribution of getting tails if you flip a coin three times:

• The table intersections represent an X–Y combination; the probabilities in those cells represent the joint probabilities.
• The probabilities in all the cells add up to 1 (because this is a probability distribution, all the probabilities must add up to 1) [1].
•  Add probabilities across rows to get the probability distribution of random variable X (also called the marginal distribution of X).
• Add probabilities down columns to get the probability distribution of random variable Y (also called the marginal distribution of Y). 

Real Life Uses

Combining two variables in a single distribution, bivariate distributions are quite common both within and outside of the world of mathematics. For example, in medical checkups, cholesterol and triglyceride levels can be combined to assess heart health; gamblers may look at pairing sixes when rolling dice; cafés could try correlating coffee sales with those for cakes; while researchers investigate how alcohol plays into car crashes. Depending on which variables you choose to study however, each bivariate distribution will present unique insights – no unified representation is possible!

Bivariate Normal Distribution

A bivariate normal distribution describes the joint probability distribution of two normally distributed variables (X, Y). Five parameters describe this distribution:

• m_x, m_y = mean values of X and Y;
• s_x, s_y = standard deviation of X and Y;
• r_xy = correlation coefficient between X and Y.

Zero correlation (r = 0) means that X and Y are independent random variables.

Three important characteristics of a bivariate distribution

Direction, shape, and strength are three important characteristics of a bivariate distribution. While we can formally define correlation between variables with a correlation coefficient, a scatterplot can give us a rough idea of how much the variables are correlated — or if they bear no relationship at all.

• Direction: does the scatterplot trend upwards (a positive linear association), or downwards (a negative linear association)?
• Shape: does the scatterplot have a shape of some sort, such as an ellipse? If so, there may be correlation. A random pattern with no discernible shape would indicate no relationship between variables.
• Strength: Is the scatterplot long and thin, indicating a strong relationship, or thick and wide, indicating a weak relationship?

See also: Marshall-Olkin Bivariate Distribution

References

[1] Gehrman, J. 6. Bivariate Rand. Vars. Bivariate Probability Distributions. Retrieved July 27, 2023 from: https://www.csus.edu/indiv/j/jgehrman/courses/stat50/bivariate/6bivarrvs.htm

[2] Liu, M. STA 611: Introduction to Mathematical Statistics; Lecture 4: Random Variables and Distributions. Retrieved July 27, 2023 from: http://www2.stat.duke.edu/courses/Fall18/sta611.01/Lecture/Lecture04.pdf

[3] Geek3, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons